Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the parametric equations of the line through the given pair of points. \((5,-3,-3),(5,4,2)\)

Step-by-Step Solution

Answer

The parametric equations are: \(x = 5\), \(y = -3 + 7t\), \(z = -3 + 5t\).

Step 1: Determine the Direction Vector

First, find the direction vector of the line by subtracting the coordinates of the first point from the coordinates of the second point. Given points are \((5,-3,-3)\) and \((5,4,2)\). The direction vector is \((5-5, 4-(-3), 2-(-3)) = (0, 7, 5)\).

Step 2: Write Parametric Equations

Using the direction vector \((0, 7, 5)\) and one of the points, say \((5,-3,-3)\), we form the parametric equations of the line: \[\begin{align*}x &= 5, \y &= -3 + 7t, \z &= -3 + 5t.\end{align*}\]This represents the line in parametric form where \(t\) is the parameter.

Key Concepts

Direction Vector

In coordinate geometry, a direction vector is essential to understanding the orientation of a line in space. It is derived by subtracting the coordinates of two points that lie on the line.
This vector indicates both the direction and the rate at which we travel from one point to another.

For example, given points \(5, -3, -3\) and \(5, 4, 2\), we can compute the direction vector as follows:

• Subtract the x-coordinates: \((5-5) = 0\)
• Subtract the y-coordinates: \((4 - (-3)) = 7\)
• Subtract the z-coordinates: \((2 - (-3)) = 5\)

Thus, the direction vector is \(0, 7, 5\).
This vector serves as a guide to translate the line within the three-dimensional space.

Parametric Form

The parametric form of a line gives us a powerful way to represent lines using equations that incorporate a parameter, typically denoted as \(t\).
By expressing the coordinates \(x, y, z\) in terms of \(t\), we can easily trace the path of the line.

In the problem, the parametric equations based on the direction vector \(0, 7, 5\) and the point \(5, -3, -3\) are:

• \(x = 5\)
• \(y = -3 + 7t\)
• \(z = -3 + 5t\)

These equations illustrate how the line moves in 3D space as \(t\) varies.
At \(t = 0\), the line is at the starting point \(5, -3, -3\); as \(t\) increases or decreases, the line advances in the direction of the vector.

Lines in 3D

Understanding lines in 3D involves visualizing how they extend through space beyond mere flat surfaces.
While in two dimensions, a line only needs direction and position, in 3D, it also requires depth.

A line in 3D can be described by vectors and points since vectors efficiently capture both direction and extent, encompassing the z-axis as well.
In our exercise, the line determined by the points \(5, -3, -3\) and \(5, 4, 2\) spans through three axes, represented by the parametric equations:

• \(x = 5\)
• \(y = -3 + 7t\)
• \(z = -3 + 5t\)

By setting various values for \(t\), we can map out every possible position the line occupies within a 3D space.

Coordinate Geometry

Coordinate geometry, or analytic geometry, bridges algebra with geometry to analyze geometric shapes using a coordinate system.
It allows us to represent lines, curves, and shapes numerically, providing a precise way to work with equations describing these forms.

In 3D coordinate geometry, a line can be defined using parametric equations derived from points and direction vectors.
This approach simplifies calculations and aids in solving complex problems related to distances, intersections, and angles.
By using direction vectors and parametric forms, we're able to model and manipulate objects effectively within the Cartesian coordinate system.
This principle was utilized in solving the problem of finding the parametric equations of the line through two given points.